Periodic Function Review Vocabulary: Periodic Function Cycle Period Midline Amplitude = 1 2 (max. – min.) Degrees vs. Radians ⋅ 𝜋 180° Degrees = Radians ⋅ 180° 𝜋 Radians = Degrees

𝒚= 𝐬𝐢𝐧 𝒙 𝒚= 𝐜𝐨𝐬 𝒙 𝒚= 𝐭𝐚𝐧 𝒙 𝑦= 𝒂 sin 𝒃(𝑥−𝒉) +𝒌 D: All Reals R: −𝟏≤𝒚≤𝟏 = 𝐬𝐢𝐧 𝒙 𝐜𝐨𝐬 𝒙 D: All Reals R: −𝟏≤𝒚≤𝟏 Period: 𝟐𝝅 D: All Reals R: −𝟏≤𝒚≤𝟏 Period: 𝟐𝝅 D: All Reals but odd values of 𝝅 𝟐 R: All reals Period: 𝝅 𝑦= 𝒂 sin 𝒃(𝑥−𝒉) +𝒌 𝑎 = Amplitude 𝑏= frequency ℎ= Phase shift 𝑘= Vertical shift Reciprocal Functions 𝐜𝐬𝐜 𝜽 = 𝟏 𝐬𝐢𝐧 𝜽 𝐬𝐞𝒄 𝜽 = 𝟏 𝐜𝐨𝐬 𝜽 𝐜𝐨𝐭 𝜽 = 𝟏 tan 𝜽 = 𝐜𝐨𝐬 𝜽 𝐬𝐢𝐧 𝜽

Periodic Functions Review Unit Circle Equation Parts Equations from Graphs Graphing: sin and cos tan, sec, csc 10 20 30 40 50

Find the exact value of: sin 23𝜋 6 Answer

sin 23𝜋 6 =− 1 2

Find the exact value of: Answer

c𝑜𝑠 − 17𝜋 4 = 2 2

Find the exact value of: 𝑡𝑎𝑛 𝜋 6 Answer

𝑡𝑎𝑛 𝜋 6 = sin 𝜋 6 cos 𝜋 6 = 1 2 3 2 = 3 3 = 1 2 ⋅ 2 3

Find the exact value of: sec 5𝜋 6 Answer

sec 5𝜋 6 = 1 cos 5𝜋 6 = −2 3 3 = 1 − 3 2

Find the exact value of: csc 5𝜋 4 Answer

csc 5𝜋 4 = 1 sin 5𝜋 4 = 1 − 2 2 =− 2

𝑦=2 sin 3𝑥 +1 Identify the following Amplitude: Phase shift: Period: Vertical Shift: Range: Answer

𝑦=2 sin 3𝑥 +1 Identify the following Amplitude: Phase shift: 2 Period: Vertical Shift: Range: 2 None or 0 2𝜋 3 Up 1 −1≤𝑦≤3

𝑦=−3 sin 2 𝑥− 𝜋 2 −5 Identify the following Amplitude: Phase shift: Period: Vertical Shift: Range: Answer

𝑦=−3 sin 2 𝑥− 𝜋 2 −5 Identify the following Amplitude: Phase shift: 3 Period: Vertical Shift: Range: 3 Right 𝜋 2 𝜋 Down 5 −8≤𝑦≤−2

𝑦=2𝜋 cos 𝜋 𝑥+ 𝜋 6 −4 Identify the following Amplitude: Phase shift: Period: Vertical Shift: Range: Answer

𝑦=2𝜋 cos 𝜋 𝑥+ 𝜋 6 −4 Identify the following Amplitude: Phase shift: 2𝜋 Period: Vertical Shift: Range: 2𝜋 Left 𝜋 6 2 Down 4 −4−2𝜋≤𝑦≤−4+2𝜋

𝑦= tan 3𝑥 −2 Identify the following Amplitude: Phase shift: Period: Vertical Shift: Range: Answer

𝑦= tan 3𝑥 −2 Identify the following Amplitude: Phase shift: None Period: Vertical Shift: Range: None 𝜋 3 Down 2 All Real Numbers

𝑦= 1 2 tan 𝑥+ 𝜋 4 +3 Identify the following Amplitude: Phase shift: Period: Vertical Shift: Range: Answer

𝑦= 1 2 tan 𝑥+ 𝜋 4 +3 Identify the following Amplitude: Phase shift: Period: Vertical Shift: Range: Vertical Stretch of 1 2 Left 𝜋 4 𝜋 Up 3 All Reals

Write a sine function for the graph. Answer

𝑦=2 sin 3𝑥 −4

Write a sine function for the graph. Answer

𝑦=2 sin 𝑥+ 𝜋 4 +1

Write a cosine function for the graph. Answer

𝑦=3 cos 𝑥− 𝜋 3 −1

Write a cosine function for the graph. Answer

𝑦=2 cos 𝑥− 3𝜋 4 +2

Write a sine function for the graph. Answer

𝑦=−2 sin 2 𝑥+ 𝜋 6 +3

Graph the following from 0 to 4𝜋: 𝑦= sin 𝑥 Answer

𝑦= sin 𝑥

Graph the following from 0 to 4𝜋: 𝑦= cos 𝑥 Answer

𝑦= cos 𝑥

Graph the following from 0 to 4𝜋: 𝑦= 2sin 0.5𝑥 Answer

𝑦= 2sin 0.5𝑥

Graph the following from 0 to 4𝜋: 𝑦= sin 𝑥− 𝜋 3 +1 Answer

𝑦= sin 𝑥− 𝜋 3 +1

Graph the following from 0 to 4𝜋: 𝑦= cos 𝑥+ 𝜋 4 −2 Answer

𝑦= cos 𝑥+ 𝜋 4 −2

Graph the following from 0 to 4𝜋: 𝑦= tan 𝑥 Answer

𝑦= tan 𝑥

Graph the following from 0 to 4𝜋: 𝑦= 2tan 0.5𝑥 Answer

𝑦= 2tan 0.5𝑥

Graph the following from 0 to 4𝜋: 𝑦= sec 𝑥 Answer

𝑦= sec 𝑥

Graph the following from 0 to 4𝜋: 𝑦= csc 𝑥 Answer

𝑦= csc 𝑥

Graph the following from 0 to 4𝜋: 𝑦=3 sin 2 𝑥+ 𝜋 3 −1 Answer

𝑦=3 sin 2 𝑥+ 𝜋 3 −1